Solution
= Solution
Intersect a <connected component> $U$ of $\rho_A(x)$ with the <clopen> subset $\rho_B(x)$. <Connectedness> makes the intersection either empty or all of $U$. In the empty case $U\subseteq\sigma_B(x)$. In the other case $U\subseteq\rho_B(x)$; a connected set in $\rho_B(x)$ containing $U$ is also connected in $\rho_A(x)$, so maximality prevents it from being larger. Thus \b[$U$ is a component of $\rho_B(x)$, or $U\subseteq\sigma_B(x)$].